The E-step computes the posterior probabilities associated with each observation using eq. (REF ); in addition to the current estimates of \({R}\)  and \({\Sigma }\) , one needs to specify the uniform-component parameter \(\kappa \) . With the explained choice for \(\pi _{in}\)  and \(\pi _{out}\)  we have \(\kappa \propto (\det {\Sigma })^{1/2}\) . The M-step estimates the transformation \({R}\)  and the covariance \({\Sigma }\) . The estimation of \({R}\)  is done in closed form using an extension of [1]} to deal with orthogonal matrices of arbitrary dimension rather than with 3-D rotations. The covariance matrix can be easily estimated from eq. (REF ) as is classically done [2]}. At convergence, EM provides an optimal value for the maximum likelihood, as well as final estimates for the posterior probabilities. This posteriors assign each “mean” observation \( \overline{x}_m\)  (eq. REF ) to a “cluster center” \(y_m\) .
